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**Chris Hillman** · Sep20-07, 08:25 PM

The theory of linear representations of finite groups was initiated c. 1896 by Georg Frobenius http://www-history.mcs.st-and.ac.uk/...Frobenius.html, who had earlier spent years working on the theory of permutation representations of finite groups. (In the first subject, given a finite group we study an isomorphic group which consists of linear transformations on some finite dimensional vector space; in the second, we study an isomorphic group which consists of permutations of some finite set. Or, sometimes, a quotient group.)

If you know about the wonderful properties of the character table of a finite group, I urge you to take the time to learn about the analogous concept for permutation groups, the so-called table of Marks. This was introduced to math students by William Burnside, Theory of Groups of Finite Order, Cambridge University Press, 1897. In his forward, Burnside http://www-groups.dcs.st-and.ac.uk/~.../Burnside.html remarked that he had chosen not to cover the Greatest New Thing, namely Frobenius's theory of linear representations, because he didn't know of anything which that theory could do which couldn't be done with permutation representations. But he soon used the new theory to prove something important and became a passionate convert, as he explained in the forward to the second edition (1911) of his textbook, which did cover linear representations. So between 1898 and 1911 representation theory passed from being a promising innovation to an essential core topic.

Incidently, legend has it that Burnside learned about the "table of Marks" from a brilliant amateur he met in the British Museum, who happened to know a German mathematician named Engel http://www-gap.dcs.st-and.ac.uk/~his...ies/Engel.html who knew Frobenius and Lie http://www-gap.dcs.st-and.ac.uk/~his...phies/Lie.html. But this man had a heavy accent, so Burnside misheard "Marx" and we wound up with the spectacularly misnamed "table of Marks", instead of "Frobenius table". Later, an Irish writer living in Paris, who had heard this story from a friend and was much amused by the collision of Scottish and German accents, tossed the phrase "three quarks for muster Mark" into his latest novel, which later inspired Gell-Mann... but I digress. My point was that while I am a huge fan of the table of Marks, it lacks the orthogonality properties which make the character table so useful for enumerating useful information about representations.

(Burnside's textbook was responsible for another famous misnomer. He stated and proved a lemma which was well known to German and French mathematicians of the day, and which was due in part to Lagrange and in part to Frobenius. But some mathematicians mistakenly attributed it to Burnside, so for a long time this result was incorrectly known in the English language literature as the Burnside lemma. And it doesn't end there: the Burnside lemma, or Cauchy-Frobenius lemma as it is now known, is needed for Polya enumeration theory, which was indeed independently discovered by Polya http://www-gap.dcs.st-and.ac.uk/~his...ies/Polya.html but which later turned out to have been earlier published by a forgotten American mathematician, Redfield.)

Pretty soon mathematicians like Hermann Weyl http://www-groups.dcs.st-and.ac.uk/~...hies/Weyl.html jumped in and initiated the analogous theory of linear representations of Lie groups, especially the so-called classical Lie groups. Deep and beautiful connections soon came to light between the theory of representations of

and

, and pretty soon there were highly developed and closely interrelated theories of invariants of, (linear) representations of, and harmonic analysis on these groups. Coxeter http://www-history.mcs.st-and.ac.uk/...s/Coxeter.html and Dynkin http://www-history.mcs.st-andrews.ac...es/Dynkin.html returned the favor to Lie theory by giving a beautiful proof of the earlier classification by Cartan http://www-groups.dcs.st-and.ac.uk/~...es/Cartan.html of the finite dimensional complex simple Lie algebras. (Hmm... "complex simple Lie algebra" sounds funny if you don't how mathematicians parse this, doesn't it?!)

Many of the great mathematicians of the twentieth century made landmark contributions to the development of representation theory, including
Issai Schur http://www-history.mcs.st-and.ac.uk/...ies/Schur.html, Richard Brauer http://www-history.mcs.st-and.ac.uk/...es/Brauer.html, Harish-Chandra http://www-groups.dcs.st-and.ac.uk/~...h-Chandra.html, Armand Borel http://www-groups.dcs.st-andrews.ac....el_Armand.html, George W. Mackey http://www-groups.dcs.st-and.ac.uk/~...es/Mackey.html,
and via harmonic analysis, one might even include Andre Weil http://www-history.mcs.st-andrews.ac...hies/Weil.html.

The multivolume book by Curtis and Reiner http://www-gap.dcs.st-and.ac.uk/~his...ly/Reiner.html contains much historical background, and a quick glance at this book (which covers basic representation theory up to the 1960s or so) should answer your question about whether this is a big subject. Or if not, well, my local research library has 152 books on representation theory, so this is a monster of a big subject! Whether you want to understand the "gauge symmetries" of some gauge theory in physics, or to study the distribution of polarized sky light (for which you need tensor spherical harmonics), you need to know about representations (particularly the basic "building blocks" of the theory, the irreducible representations, or irreps for short) of the appropriate group. And don't even get me started on ergodic theory!

There are indeed deep connections with topology and analysis (and with much more besides, such as Kleinian geometry and Cartanian geometry--- the latter subject, the common generalization of Kleinian geometry and Riemannian geometry, is currently undergoing an impressive revival.). I already hinted at a connection with harmonic analysis (a vast generalization of the theory of Fourier transforms). As for a connection with topology, I might mention the theory of covering spaces, which is closely connected to homotopy groups in algebraic topology, and I might mention the concept of the universal covering group, a simply connected Lie group which plays an important role in much modern mathematics (among many other things, this notion provides some important examples of Lie groups which are not realizable as matrix groups). And there are indeed important connections with modules. Indeed, when R is the group ring of our group, R-modules--- oh never mind, I've said enough!

For evidence that the theory continues to grow at a daunting rate, see http://www.arxiv.org/list/math.RT/recent

I didn't see Matt Grimes's comment but I'll go out on a limb and guess that he was thinking of representations of categories, a subject which category theorists consider to be "concrete" but which many others might term "abstract", if not nonsense. Or he might have been referring to something like diagram chasing in homological algebra, a technique which certainly might have come up while discussing the cohomology of Lie groups.

Classic references for one man's viewpoint:

George W. Mackey, The Scope And History Of Commutative And Noncommutative Harmonic Analysis, American Mathematical Society, 1992.

George W. Mackey, The theory of group representations, University of Chicago Press, 1955.

George W. Mackey, Unitary group representations in physics, probability, and number theory, Benjamin/Cummings, 1978.

C. C. Moore, editor. Group representations, ergodic theory, operator algebras, and mathematical physics: proceedings of a conference in honor of George W. Mackey.
Springer, 1987.

Also relevant:

Charles W. Curtis, Pioneers of Representation Theory, Am. Math. Soc., 1999. (Same Curtis as in Curtis and Reiner.) See also this book review: http://www.ams.org/bull/2000-37-03/S...00-00867-3.pdf by J. E. Humphreys, author of one of many textbooks on representation theory.

(Edit: after having written the above, I discovered that this review is on-line and not surprisingly it is a much better summary of the history of representation theory than what I came up with above.)

Er... did I answer the question? Or at least clarify why it might hard to concisely answer the question?

Sep20-07, 08:25 PM	Last edited by Chris Hillman; Sep20-07 at 10:19 PM.. Reason: delete name dropping #4
Chris Hillman Chris Hillman is Offline: Posts: 1,862 Recognitions: Science Advisor	Tiny sketch of history and scope of representation theory The theory of linear representations of finite groups was initiated c. 1896 by Georg Frobenius http://www-history.mcs.st-and.ac.uk/...Frobenius.html, who had earlier spent years working on the theory of permutation representations of finite groups. (In the first subject, given a finite group we study an isomorphic group which consists of linear transformations on some finite dimensional vector space; in the second, we study an isomorphic group which consists of permutations of some finite set. Or, sometimes, a quotient group.) If you know about the wonderful properties of the character table of a finite group, I urge you to take the time to learn about the analogous concept for permutation groups, the so-called table of Marks. This was introduced to math students by William Burnside, Theory of Groups of Finite Order, Cambridge University Press, 1897. In his forward, Burnside http://www-groups.dcs.st-and.ac.uk/~.../Burnside.html remarked that he had chosen not to cover the Greatest New Thing, namely Frobenius's theory of linear representations, because he didn't know of anything which that theory could do which couldn't be done with permutation representations. But he soon used the new theory to prove something important and became a passionate convert, as he explained in the forward to the second edition (1911) of his textbook, which did cover linear representations. So between 1898 and 1911 representation theory passed from being a promising innovation to an essential core topic. Incidently, legend has it that Burnside learned about the "table of Marks" from a brilliant amateur he met in the British Museum, who happened to know a German mathematician named Engel http://www-gap.dcs.st-and.ac.uk/~his...ies/Engel.html who knew Frobenius and Lie http://www-gap.dcs.st-and.ac.uk/~his...phies/Lie.html. But this man had a heavy accent, so Burnside misheard "Marx" and we wound up with the spectacularly misnamed "table of Marks", instead of "Frobenius table". Later, an Irish writer living in Paris, who had heard this story from a friend and was much amused by the collision of Scottish and German accents, tossed the phrase "three quarks for muster Mark" into his latest novel, which later inspired Gell-Mann... but I digress. My point was that while I am a huge fan of the table of Marks, it lacks the orthogonality properties which make the character table so useful for enumerating useful information about representations. (Burnside's textbook was responsible for another famous misnomer. He stated and proved a lemma which was well known to German and French mathematicians of the day, and which was due in part to Lagrange and in part to Frobenius. But some mathematicians mistakenly attributed it to Burnside, so for a long time this result was incorrectly known in the English language literature as the Burnside lemma. And it doesn't end there: the Burnside lemma, or Cauchy-Frobenius lemma as it is now known, is needed for Polya enumeration theory, which was indeed independently discovered by Polya http://www-gap.dcs.st-and.ac.uk/~his...ies/Polya.html but which later turned out to have been earlier published by a forgotten American mathematician, Redfield.) Pretty soon mathematicians like Hermann Weyl http://www-groups.dcs.st-and.ac.uk/~...hies/Weyl.html jumped in and initiated the analogous theory of linear representations of Lie groups, especially the so-called classical Lie groups. Deep and beautiful connections soon came to light between the theory of representations of and , and pretty soon there were highly developed and closely interrelated theories of invariants of, (linear) representations of, and harmonic analysis on these groups. Coxeter http://www-history.mcs.st-and.ac.uk/...s/Coxeter.html and Dynkin http://www-history.mcs.st-andrews.ac...es/Dynkin.html returned the favor to Lie theory by giving a beautiful proof of the earlier classification by Cartan http://www-groups.dcs.st-and.ac.uk/~...es/Cartan.html of the finite dimensional complex simple Lie algebras. (Hmm... "complex simple Lie algebra" sounds funny if you don't how mathematicians parse this, doesn't it?!) Many of the great mathematicians of the twentieth century made landmark contributions to the development of representation theory, including Issai Schur http://www-history.mcs.st-and.ac.uk/...ies/Schur.html, Richard Brauer http://www-history.mcs.st-and.ac.uk/...es/Brauer.html, Harish-Chandra http://www-groups.dcs.st-and.ac.uk/~...h-Chandra.html, Armand Borel http://www-groups.dcs.st-andrews.ac....el_Armand.html, George W. Mackey http://www-groups.dcs.st-and.ac.uk/~...es/Mackey.html, and via harmonic analysis, one might even include Andre Weil http://www-history.mcs.st-andrews.ac...hies/Weil.html. The multivolume book by Curtis and Reiner http://www-gap.dcs.st-and.ac.uk/~his...ly/Reiner.html contains much historical background, and a quick glance at this book (which covers basic representation theory up to the 1960s or so) should answer your question about whether this is a big subject. Or if not, well, my local research library has 152 books on representation theory, so this is a monster of a big subject! Whether you want to understand the "gauge symmetries" of some gauge theory in physics, or to study the distribution of polarized sky light (for which you need tensor spherical harmonics), you need to know about representations (particularly the basic "building blocks" of the theory, the irreducible representations, or irreps for short) of the appropriate group. And don't even get me started on ergodic theory! There are indeed deep connections with topology and analysis (and with much more besides, such as Kleinian geometry and Cartanian geometry--- the latter subject, the common generalization of Kleinian geometry and Riemannian geometry, is currently undergoing an impressive revival.). I already hinted at a connection with harmonic analysis (a vast generalization of the theory of Fourier transforms). As for a connection with topology, I might mention the theory of covering spaces, which is closely connected to homotopy groups in algebraic topology, and I might mention the concept of the universal covering group, a simply connected Lie group which plays an important role in much modern mathematics (among many other things, this notion provides some important examples of Lie groups which are not realizable as matrix groups). And there are indeed important connections with modules. Indeed, when R is the group ring of our group, R-modules--- oh never mind, I've said enough! For evidence that the theory continues to grow at a daunting rate, see http://www.arxiv.org/list/math.RT/recent I didn't see Matt Grimes's comment but I'll go out on a limb and guess that he was thinking of representations of categories, a subject which category theorists consider to be "concrete" but which many others might term "abstract", if not nonsense. Or he might have been referring to something like diagram chasing in homological algebra, a technique which certainly might have come up while discussing the cohomology of Lie groups. Classic references for one man's viewpoint: George W. Mackey, The Scope And History Of Commutative And Noncommutative Harmonic Analysis, American Mathematical Society, 1992. George W. Mackey, The theory of group representations, University of Chicago Press, 1955. George W. Mackey, Unitary group representations in physics, probability, and number theory, Benjamin/Cummings, 1978. C. C. Moore, editor. Group representations, ergodic theory, operator algebras, and mathematical physics: proceedings of a conference in honor of George W. Mackey. Springer, 1987. Also relevant: Charles W. Curtis, Pioneers of Representation Theory, Am. Math. Soc., 1999. (Same Curtis as in Curtis and Reiner.) See also this book review: http://www.ams.org/bull/2000-37-03/S...00-00867-3.pdf by J. E. Humphreys, author of one of many textbooks on representation theory. (Edit: after having written the above, I discovered that this review is on-line and not surprisingly it is a much better summary of the history of representation theory than what I came up with above.) Er... did I answer the question? Or at least clarify why it might hard to concisely answer the question?